THE UNIVERSITY OF ZAMBIA
POST GRADUATE SCHOOL OF BUSINESS
NOVEMBER MOCK EXAMINATIONS
MARKETING/BPS/BBA / BACC 1300-MATHEMATICAL ANALYSIS
Date: 06 /11/2022
Time: 09:00hrs TIME
ALLOWED: FOUR (4) HOURS
INSTRUCTIONS TO CANDIDATES
1. There are SEVEN (7) questions in this paper.
2. Attempt 𝐀𝐋𝐋 showing all necessary working
3. Use of 𝐓𝐈𝐏𝐄𝐗 is NOT allowed
DO NOT TURN THIS PAPER UNTIL YOU ARE TOLD TO DO SO
QUESTION ONE
A triple A battery manufacturer determines that at a price of K50, 50 units of the triple A
batteries are sold, further analysis showed that at a price of K60, 40 units are sold.
The manufacturer also determines that the fixed costs of producing x units of the triple A type of
batteries is 300 and the variable costs are 20.
a. Determine
i.
The price function in terms p = mx
[4]
ii.
The total revenue function "R(x)"
[2]
iii. The total cost function C(x)
[1]
iv.
Find the profit function "π(x)"
[2]
b. Under what conditions will the firm break-even
[1]
c. Determine algebraically at break-even for quantity
i.
The total revenue
[3]
ii.
The total cost
[3]
d. On the same graph sketch the R(x) and C(x), indicating clearly the break-even points [7]
e. From "d", giving a reason for your answer, which break-even values are feasible for
business sustenance,
[2]
f. Using your result in "e" at what price should the batteries be sold at to break-even as the
case of "d"
[1]
g. On a separate graph, search the profit function π(x)
[5]
h. From your graph "f" , how many units maximizes the revenues and minimizes the costs [2]
i. From your graph "g", how many units maximizes the profits
[1]
j. What is the maximum revenue and profit and minimum costs in the case of g" and "f" [2]
k. When revenue is maximized, determine
i. The selling price of the triple A battery
[1]
ii. The cost of production
[1]
l. When profits are maximized, determine
i.
The selling price of the triple A battery
[1]
ii.
The cost of production
[1]
[𝐓𝐎𝐓𝐀𝐋 𝟒𝟎]
QUESTION TWO
A screening test for a particular disease is applied to everyone in a large population. The test
classifies people into three groups: ‘positive’, ‘doubtful’ and ‘negative’. Of the population, 3% is
classified as positive, 6% as doubtful and the rest negative.
In fact, of the people who test positive, only 95% have the disease. Of the people who test
doubtful, 10% have the disease. Of the people who test negative, 1% actually have the disease.
People who do not have the disease are described as ‘clear’.
a. Copy and complete the tree diagram to show this information.
[5]
b. Find the probability that a randomly selected person tests negative and is clear.
c. Find the probability that a randomly selected person has the disease.
d. Find the probability that a randomly selected person tests negative given that the
person has the disease.
e. Comment briefly on what your answer to part (d) indicates about the effectiveness
of the screening test.
[2]
[3]
[3]
[2]
Once the test has been carried out, those people who test doubtful are given a detailed
medical examination. If a person has the disease the examination will correctly identify
this in 98% of cases.
If a person is clear, the examination will always correctly identify this.
f. A person is selected at random. Find the probability that this person either tests negative
originally or tests doubtful and is then cleared in the detailed medical examination.
[5]
[𝐓𝐎𝐓𝐀𝐋 𝟐𝟎]
QUESTION THREE
MARKS
Traffic engineers are studying the correlation between traffic flow on a busy main road
and air pollution at a nearby air quality monitoring station. Traffic flow is recorded
automatically by sensors and reported each hour as the average flow in vehicles per hour
for the preceding hour. The air quality monitoring station provides, each hour, an overall
pollution reading in a suitable unit (higher readings indicate more pollution).
Data for a random sample of 15 hours are as follows
TRAFIC FLW
POLLUTION
1815 2206 1835 1918 2420 2315 1796
3.5
8.3
5.0
4.8
20.0 18.0 3.6
TRAFIC FLW
POLLUTION
2588 2040 2368 2170 2285 2120
24.4 9.5
16.2 10.6 13.8 12.0
2850
32.0
2635
24.2
a. State with a reason which one is the explanatory variable.
[1]
b. Draw a scatter diagram to illustrate these data and comment on it briefly
[2]
c. Calculate Product moment correlation coefficient, and interpret value in the context
of the question.
[5]
d. Calculate the coefficient of determination and interpret in the context of this question [2]
e. Determine the probable error of the result in "𝑏" above and interpret in the context of
this question.
[4]
[5]
f. Calculate the equation of the least squares regression line.
g. Estimate the air pollution recorded at the nearby air quality monitoring station when the
traffic recorded automatically by the censors is 2022
[1]
[𝐓𝐎𝐓𝐀𝐋 𝟐𝟎]
QUESTION FOUR
a. Given that 2lnx + ln3 = ln(5x + 2). Show that it can be written in the
form px 2 + qx + r = 0 , stating the values of p, q and r
b. Hence, show that 2lnx + ln3 = ln(5x + 2) has only one valid solution.
c. Given f(x) = 2x 2 + 6x + 3,
i.
Use first principles to find f ′′ (x)
ii.
Hence find f ′ (3)
MARKS
[4]
[5]
[8]
[2]
[𝐓𝐎𝐓𝐀𝐋 𝟐𝟎]
QUESTION FIVE
MARKS
Brian supplies customized ear buds. At a price of 𝐾6, and 𝐾12, only 12 and 5
are demanded respectively, at a price of 𝐾16 𝑎𝑛𝑑 𝐾12 only 13 𝑎𝑛𝑑 9 supplied respectively.
a. Write the supply and demand functions in the form of 𝑞 = 𝑚𝑝
b. Define the following terms
i.
Equilibrium levels
ii.
Consumer surplus
iii.
Producer surplus
iv.
Total surplus
c. Hence, algebraically determine the equilibrium prices and quantities
𝑎
(leaving your answer in the form 𝑏 , 𝑏 > 0)
[3]
[1]
[1]
[1]
[1]
[4]
d. On the same graph, sketch the demand and supply functions explicitly showing
equilibrium quantities, the supply and quantity intercepts, the demand and
consumer surplus.
e. From your graph calculate
i.
The consumer surplus
ii.
Producer surplus
iii.
Total surplus
[4]
[2]
[2]
[1]
[𝐓𝐎𝐓𝐀𝐋 𝟐𝟎]
QUESTION SIX
MARKS
38
a. Given 10 + 7√5 + 1−2√5 = 𝑎 + 𝑏√𝑑 find the values of "𝑎" , "𝑏", 𝑎𝑛𝑑 "𝑑"
̅̅̅̅̅ = 𝑝
b. Find the integer values of "𝑝" and "𝑞" given 12.145123
c. Given 𝑓(𝑥) = 2𝑥 3 + 6𝑥 2 + 3,
𝑞
[4]
[4]
i.
Use first principles to find 𝑓 ′′ (𝑥)
[10]
ii.
Hence find 𝑓 ′ (3)
[2]
[𝐓𝐎𝐓𝐀𝐋 𝟐𝟎]
𝐌𝐀𝐑𝐊𝐒
QUESTION SEVEN
Given the revenue function
𝑥2
𝑅(𝑥) = √𝑥 2
+5
a. Find the approximate marginal revenue function
b. Find in simplified form the exact marginal revenue functional
c. At a market level of 12 units per day
i. Use your result in "𝑎" to find the approximate marginal revenue
ii. Use your result in "b" to find the exact marginal revenue
[𝟓]
[7]
[3]
[5]
[𝐓𝐎𝐓𝐀𝐋 𝟐𝟎]